Geometry optimization
We use DFTK and the GeometryOptimization package to find the minimal-energy bond length of the $H_2$ molecule. First we set up an appropriate DFTKCalculator (see AtomsCalculators integration), for which we use the LDA model just like in the Tutorial for silicon.
using DFTK
calc = DFTKCalculator(;
model_kwargs = (; functionals=LDA()), # model_DFT keyword arguments
basis_kwargs = (; kgrid=[1, 1, 1], Ecut=10) # PlaneWaveBasis keyword arguments
)DFTKCalculator{Nothing}(DFTK.DFTKParameters((functionals = Xc(lda_x, lda_c_pw),), (kgrid = [1, 1, 1], Ecut = 10), (callback = identity,)), nothing, Base.RefValue{Int64}(0), Base.RefValue{Int64}(0), true)Next we set up an initial hydrogen molecule within a box of vacuum. We use the parameters of the equivalent tutorial from ABINIT, that is a simulation box of 10 bohr times 10 bohr times 10 bohr and a pseudodojo pseudopotential.
using LinearAlgebra
using LazyArtifacts
r0 = 1.4 # Initial bond length in Bohr
a = 10.0 # Box size in Bohr
lattice = a * I(3)
H = ElementPsp(:H; psp=load_psp(artifact"pd_nc_sr_pbe_standard_0.4.1_upf/H.upf"));
atoms = [H, H]
positions = [zeros(3), lattice \ [r0, 0., 0.]]
h2_crude = periodic_system(lattice, atoms, positions)FlexibleSystem(H₂, periodic = TTT):
bounding_box : [ 10 0 0;
0 10 0;
0 0 10]u"a₀"
Atom(H, [ 0, 0, 0]u"a₀")
Atom(H, [ 1.4, 0, 0]u"a₀")
.------------.
/| |
/ | |
* | |
| | |
| | |
| .------------.
| / /
|/ /
H-H----------*
Finally we call minimize_energy! to start the geometry optimisation. We use verbosity=2 to get some insight into the minimisation. With verbosity=1 only a summarising table would be printed and with verbosity=0 (default) the minimisation would be quiet.
using GeometryOptimization
results = minimize_energy!(h2_crude, calc; tol_forces=2e-6, verbosity=2)n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.110556589431 -0.82 0.80 8.0 27.0ms
2 -1.117202297379 -2.18 -1.86 0.80 1.0 17.3ms
3 -1.117251826676 -4.31 -2.71 0.80 1.0 17.0ms
4 -1.117252479208 -6.19 -3.55 0.80 1.0 17.5ms
5 -1.117252499101 -7.70 -3.99 0.80 1.0 17.9ms
6 -1.117252500639 -8.81 -4.54 0.80 1.0 18.3ms
7 -1.117252500840 -9.70 -5.40 0.80 1.0 35.0ms
8 -1.117252500846 -11.19 -6.24 0.80 2.0 21.7ms
9 -1.117252500846 -12.90 -6.82 0.80 1.0 19.3ms
10 -1.117252500846 -14.51 -6.98 0.80 2.0 21.8ms
11 -1.117252500846 -14.37 -8.36 0.80 1.0 19.6ms
12 -1.117252500846 -15.05 -8.29 0.80 2.0 21.9ms
13 -1.117252500846 + -15.35 -8.74 0.80 1.0 20.1ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.117252500846 -9.26 0.80 1.0 79.8ms
2 -1.117252500846 -14.75 -10.31 0.80 1.0 16.9ms
Geometry optimisation convergence (in atomic units)
┌─────┬─────────────────┬───────────┬────────────┬────────┐
│ n │ Energy │ log10(ΔE) │ max(Force) │ Δtime │
├─────┼─────────────────┼───────────┼────────────┼────────┤
│ 0 │ -1.117252500846 │ │ 0.0269316 │ 2.27s │
└─────┴─────────────────┴───────────┴────────────┴────────┘
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.117519305142 -2.46 0.80 1.0 10.9ms
2 -1.117521893101 -5.59 -3.49 0.80 1.0 17.1ms
3 -1.117521920034 -7.57 -4.23 0.80 1.0 17.0ms
4 -1.117521920498 -9.33 -5.14 0.80 1.0 17.8ms
5 -1.117521920514 -10.81 -5.92 0.80 1.0 18.1ms
6 -1.117521920514 -12.13 -6.42 0.80 1.0 18.0ms
7 -1.117521920514 -13.35 -7.39 0.80 1.0 19.2ms
8 -1.117521920514 + -14.70 -8.18 0.80 1.0 19.3ms
9 -1.117521920514 -14.61 -8.97 0.80 1.0 19.5ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118248561944 -1.69 0.80 1.0 11.0ms
2 -1.118340340104 -4.04 -2.70 0.80 1.0 37.6ms
3 -1.118341307269 -6.01 -3.44 0.80 1.0 17.4ms
4 -1.118341324709 -7.76 -4.36 0.80 1.0 17.7ms
5 -1.118341325294 -9.23 -5.12 0.80 1.0 18.0ms
6 -1.118341325321 -10.57 -5.62 0.80 1.0 18.4ms
7 -1.118341325323 -11.73 -6.68 0.80 1.0 18.9ms
8 -1.118341325323 -13.53 -7.58 0.80 1.0 19.5ms
9 -1.118341325323 -14.95 -8.12 0.80 1.0 19.5ms
10 -1.118341325323 + -14.65 -8.55 0.80 1.0 19.4ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118341325323 -9.05 0.80 1.0 10.9ms
2 -1.118341325323 + -15.35 -10.36 0.80 1.0 17.2ms
┌─────┬─────────────────┬───────────┬────────────┬────────┐
│ n │ Energy │ log10(ΔE) │ max(Force) │ Δtime │
├─────┼─────────────────┼───────────┼────────────┼────────┤
│ 1 │ -1.118341325323 │ -2.96 │ 0.00297052 │ 1.86s │
└─────┴─────────────────┴───────────┴────────────┴────────┘
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118348017537 -3.09 0.80 1.0 15.0ms
2 -1.118348160786 -6.84 -4.11 0.80 1.0 17.0ms
3 -1.118348162263 -8.83 -4.86 0.80 1.0 17.1ms
4 -1.118348162289 -10.59 -5.78 0.80 1.0 21.6ms
5 -1.118348162289 -12.08 -6.55 0.80 1.0 25.5ms
6 -1.118348162290 -13.41 -7.04 0.80 1.0 25.6ms
7 -1.118348162290 -14.61 -8.13 0.80 1.0 22.2ms
8 -1.118348162290 -14.65 -8.95 0.80 1.0 19.0ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118356019869 -2.61 0.80 1.0 23.2ms
2 -1.118357382893 -5.87 -3.62 0.80 1.0 17.5ms
3 -1.118357396968 -7.85 -4.37 0.80 1.0 17.4ms
4 -1.118357397214 -9.61 -5.29 0.80 1.0 17.6ms
5 -1.118357397222 -11.10 -6.06 0.80 1.0 17.9ms
6 -1.118357397222 -12.45 -6.55 0.80 1.0 18.2ms
7 -1.118357397222 -13.62 -7.65 0.80 1.0 18.8ms
8 -1.118357397222 -14.40 -8.45 0.80 1.0 18.9ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118357397222 -9.24 0.80 1.0 10.9ms
2 -1.118357397222 + -14.57 -9.82 0.80 1.0 17.1ms
┌─────┬─────────────────┬───────────┬────────────┬────────┐
│ n │ Energy │ log10(ΔE) │ max(Force) │ Δtime │
├─────┼─────────────────┼───────────┼────────────┼────────┤
│ 2 │ -1.118357397222 │ -4.79 │ 4.47258e-5 │ 1.80s │
└─────┴─────────────────┴───────────┴────────────┴────────┘
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118357398684 -4.94 0.80 1.0 11.0ms
2 -1.118357398714 -10.54 -5.96 0.80 1.0 17.0ms
3 -1.118357398714 -12.52 -6.70 0.80 1.0 17.2ms
4 -1.118357398714 -14.45 -7.62 0.80 1.0 17.6ms
5 -1.118357398714 -14.51 -8.39 0.80 1.0 18.1ms
6 -1.118357398714 + -Inf -8.89 0.80 1.0 18.3ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118357400607 -4.40 0.80 1.0 10.9ms
2 -1.118357400953 -9.46 -5.42 0.80 1.0 17.2ms
3 -1.118357400956 -11.45 -6.17 0.80 1.0 17.1ms
4 -1.118357400956 -13.22 -7.09 0.80 1.0 17.5ms
5 -1.118357400956 -14.65 -7.86 0.80 1.0 17.9ms
6 -1.118357400956 -15.35 -8.35 0.80 1.0 32.6ms
7 -1.118357400956 + -Inf -9.45 0.80 1.0 18.9ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118357400956 -10.08 0.80 1.0 11.1ms
2 -1.118357400956 + -15.18 -11.02 0.80 1.0 17.1ms
┌─────┬─────────────────┬───────────┬────────────┬────────┐
│ n │ Energy │ log10(ΔE) │ max(Force) │ Δtime │
├─────┼─────────────────┼───────────┼────────────┼────────┤
│ 3 │ -1.118357400956 │ -8.43 │ 1.15243e-8 │ 1.73s │
└─────┴─────────────────┴───────────┴────────────┴────────┘Structure after optimisation (note that the atom has wrapped around)
results.systemFlexibleSystem(H₂, periodic = TTT):
bounding_box : [ 10 0 0;
0 10 0;
0 0 10]u"a₀"
Atom(H, [-0.0431826, 1.28704e-10, -2.04063e-10]u"a₀")
Atom(H, [ 1.44318, 1.27281e-10, -2.02326e-10]u"a₀")
.------------.
/| |
/ | |
* H| H |
| | |
| | |
| .------------.
| / /
|/ /
*------------*
Compute final bond length:
rmin = norm(position(results.system[1]) - position(results.system[2]))
println("Optimal bond length: ", rmin)Optimal bond length: 1.4863652613886027 a₀Our results (1.486 Bohr) agrees with the equivalent tutorial from ABINIT.